There is also another reason, more theoretical and fundamental, to attempt a 3HDM study of the BGL paradigm. In many instances, comparisons of the 2HDM with 3HDMs have revealed how special a model the 2HDM is. To give only a few examples, tree-level vacuum stability against charge breaking or spontaneous CP breaking was found for charge-and-CP conserving minima within the 2HDM [1]}, [2]}, [3]}, but charge breaking minima were found to coexist with charge-preserving ones for NHDMs with \(N\ge 3\)  [4]}; a full listing of all possible symmetries of the \({\rm SU}(2)\times {\rm U}(1)\)  invariant 2HDM was found [5]}, [6]} while for 3HDM we refer to [7]}, [8]}, [9]}; generic bounded-from-below [5]}, [6]} and unitarity [12]} bounds were found for the 2HDM, but for the 3HDM such bounds only exist for particular versions of the model. As such, the possibility of ascertaining whether the BGL structure can be extended to a full 3HDM compels us to try to find it. And of course one can obtain an exact BGL-like 3HDM – all one needs
to do is copying the procedure detailed in the previous section for the first two doublets,
while keeping the third doublet VEVless. For that to happen one would
consider the construction of the Inert Doublet Model (IDM) [13]}, [14]}, [15]}, [16]} and impose a \(\mathbb {Z}_2\)  symmetry on the third doublet, \(\phi _3 \rightarrow -\phi _3\) . This model would have FCNCs in the visible sector and also a dark matter candidate stemming from the third doublet. An interesting model would be recovered, however it would not solve our aim to
obtain a larger freedom to fit the flavour observables in comparison to what one has in the 2HDM BGL model. Thus we are compelled to consider a 3HDM wherein all doublets acquire VEVs.
